For α in (1,2) the expected 2q-moment of the normalized sum of d_α(n) f(n) is bounded by (log x)^{2q(α-1)} over a power of log log x, uniformly for q up to 1/α.
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New bounds showing that a nontrivial shift of a multiplicative subgroup containing a product set AB has |A||B| essentially bounded by |G|, plus first nontrivial upper bounds on generalized Diophantine tuples over finite fields and progress on a conjecture of Sárközy.
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Partial sums of random multiplicative functions with supercritical divisor twists
For α in (1,2) the expected 2q-moment of the normalized sum of d_α(n) f(n) is bounded by (log x)^{2q(α-1)} over a power of log log x, uniformly for q up to 1/α.
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Multiplicative structure of shifted multiplicative subgroups and its applications to Diophantine tuples
New bounds showing that a nontrivial shift of a multiplicative subgroup containing a product set AB has |A||B| essentially bounded by |G|, plus first nontrivial upper bounds on generalized Diophantine tuples over finite fields and progress on a conjecture of Sárközy.